Oscillating Brownian motion

An ‘oscillating' version of Brownian motion is defined and studied. ‘Ordinary' Brownian motion and ‘reflecting' Brownian motion are shown to arise as special cases. Transition densities, first-passage time distributions, and occupation time distributions for the process are obtained explicitly. Convergence of a simple oscillating random walk to an oscillating Brownian motion process is established by using results of Stone (1963).

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Oscillating Brownian motion

Journal of Applied Probability ◽

10.1017/s0021900200045599 ◽

1978 ◽

Vol 15 (02) ◽

pp. 300-310 ◽

Cited By ~ 3

Author(s):

Julian Keilson ◽

Jon A. Wellner

Keyword(s):

Brownian Motion ◽

Random Walk ◽

First Passage Time ◽

Passage Time ◽

Brownian Motion Process ◽

First Passage ◽

Special Cases ◽

Motion Process ◽

Transition Densities ◽

Oscillating Random Walk

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On the first-passage time of integrated Brownian motion

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/jamsa.2005.237 ◽

2005 ◽

Vol 2005 (3) ◽

pp. 237-246

Author(s):

Christian H. Hesse

Keyword(s):

Brownian Motion ◽

First Passage Time ◽

Stopping Time ◽

Passage Time ◽

Accurate Approximation ◽

Brownian Motion Process ◽

First Passage ◽

Conditional Mean ◽

Conditional Moments ◽

Motion Process

Let (Bt;t≥0) be a Brownian motion process starting from B0=ν and define Xν(t)=∫0tBsds. For a≥0, set τa,ν:=inf{t:Xν(t)=a} (with inf φ=∞). We study the conditional moments of τa,ν given τa,ν<∞. Using martingale methods, stopping-time arguments, as well as the method of dominant balance, we obtain, in particular, an asymptotic expansion for the conditional mean E(τa,ν|τa,ν<∞) as ν→∞. Through a series of simulations, it is shown that a truncation of this expansion after the first few terms provides an accurate approximation to the unknown true conditional mean even for small ν.

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Mean first-passage times of Brownian motion and related problems

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1952.0051 ◽

1952 ◽

Vol 211 (1106) ◽

pp. 431-443 ◽

Cited By ~ 42

Keyword(s):

Brownian Motion ◽

First Passage Time ◽

Passage Time ◽

First Passage Times ◽

Analytical Relationship ◽

First Passage ◽

Special Boundaries ◽

Mean First Passage Times ◽

Special Cases ◽

Passage Times

The theory of first-passage times of Brownian motion is developed in general, and it is shown that for certain special boundaries—the only ones of any importance—mean first-passage times can be derived very simply, avoiding the usual method involving series. Moreover, these formulae have a close analytical relationship to the better-known type of formulae for average 'displacements’ in given intervals; there exist certain pairs of reciprocal relations. Some new formulae, of mathematical interest, for translational Brownian motion are given. The main application of the general theory, however, lies in the derivation of experimentally particularly useful formulae for rotational Brownian motion. Special cases when external forces are present, and mean reciprocal first-passage times are discussed briefly, and finally it is shown how finite times of observation modify the mean first-passage time formulae of free Brownian motion.

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On the First Passage time for Brownian Motion Subordinated by a Lévy Process

Journal of Applied Probability ◽

10.1239/jap/1238592124 ◽

2009 ◽

Vol 46 (1) ◽

pp. 181-198 ◽

Cited By ~ 9

Author(s):

T. R. Hurd ◽

A. Kuznetsov

Keyword(s):

Brownian Motion ◽

Approximation Scheme ◽

First Passage Time ◽

Gaussian Model ◽

Passage Time ◽

Financial Mathematics ◽

First Passage ◽

Financial Modeling ◽

Motion Time ◽

Normal Inverse Gaussian

In this paper we consider the class of Lévy processes that can be written as a Brownian motion time changed by an independent Lévy subordinator. Examples in this class include the variance-gamma (VG) model, the normal-inverse Gaussian model, and other processes popular in financial modeling. The question addressed is the precise relation between the standard first passage time and an alternative notion, which we call the first passage of the second kind, as suggested by Hurd (2007) and others. We are able to prove that the standard first passage time is the almost-sure limit of iterations of the first passage of the second kind. Many different problems arising in financial mathematics are posed as first passage problems, and motivated by this fact, we are led to consider the implications of the approximation scheme for fast numerical methods for computing first passage. We find that the generic form of the iteration can be competitive with other numerical techniques. In the particular case of the VG model, the scheme can be further refined to give very fast algorithms.

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On the area under a continuous time Brownian motion till its first-passage time

Journal of Physics A General Physics ◽

10.1088/0305-4470/38/19/004 ◽

2005 ◽

Vol 38 (19) ◽

pp. 4097-4104 ◽

Cited By ~ 43

Author(s):

Michael J Kearney ◽

Satya N Majumdar

Keyword(s):

Brownian Motion ◽

Continuous Time ◽

First Passage Time ◽

Passage Time ◽

First Passage

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Occupation Times for Markov-Modulated Brownian Motion

Journal of Applied Probability ◽

10.1017/s0021900200009268 ◽

2012 ◽

Vol 49 (02) ◽

pp. 549-565 ◽

Cited By ~ 4

Author(s):

Lothar Breuer

Keyword(s):

Brownian Motion ◽

First Passage Time ◽

Passage Time ◽

Laplace Transforms ◽

Occupation Times ◽

First Passage ◽

Scale Functions ◽

Positive Variation ◽

Markov Modulated

In this paper we determine the distributions of occupation times of a Markov-modulated Brownian motion (MMBM) in separate intervals before a first passage time or an exit from an interval. We derive the distributions in terms of their Laplace transforms, and we also distinguish between occupation times in different phases. For MMBMs with strictly positive variation parameters, we further propose scale functions.

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